Shifted Laplace and related preconditioning for the Helmholtz equation

Transcription

1 Shifted Laplace and related preconditioning for the Helmholtz equation Ivan Graham and Euan Spence (Bath, UK) Collaborations with: Paul Childs (Schlumberger Gould Research), Martin Gander (Geneva) Douglas Shanks (Bath) Eero Vainikko (Tartu, Estonia) Oxford, March 2014

2 Outline of talk: seismic imaging (Schlumberger Gould Research) high-frequency Helmholtz, variable wave speed Conventional FE discretization, efficient solvers? Iterative method: GMRES convergence rates? preconditioners based on absorption brief summary of mathematical results numerical experiments Chandler-Wilde, IGG, Langdon, Spence Acta Numerica 2012: Numerical-asymptotic boundary integral methods in high-frequency scattering See also Dave Hewett s talk

3 Motivation

4 Seismic inversion Inverse problem: reconstruct material properties of subsurface (characterised by wave speed c(x)) from observed echos. Regularised iterative method: repeated solution of the (forward problem): the wave equation Frequency domain: u + 1 c 2 2 u t 2 = f or its elastic variant ( ) ωl 2 u u = f, ω = frequency c solve for u with approximate c. Large domain of characteristic length L. effectively high frequency

5 Seismic inversion Inverse problem: reconstruct material properties of subsurface (wave speed c(x)) from observed echos. Regularised iterative method: repeated solution of the (forward problem): the wave equation Frequency domain: u + 2 u t 2 = f or its elastic variant ( ) ωl 2 u u = f, ω = frequency c solve for u with approximate c. Large domain of characteristic length L. effectively high frequency

6 Marmousi Model Problem Time domain: explicit finite difference methods (slow) Frequency domain: large linear systems for each ω Solver of choice (2007) based on principle of limited absorption (Erlangga, Osterlee, Vuik, 2004)... This work: develop better solvers, (parallel algorithms?)

7 Model interior impedance problem u k 2 u = f in bounded domain Ω u iku n = g on Γ := Ω Results all hold for truncated sound-soft scattering problems in Ω (large R) B R Ω Ω Γ

8 Linear algebra problem weak form with absorption k 2 k 2 + iε, η = η(k, ε) ( a ε (u, v) := u. v (k 2 +k 2 )uv ) ik uv Ω Γ = fv + gv ShiftedLaplacian Ω Γ [Equivalently k 2 + iε (k + iρ) 2 ] finite element discretization A ε u := (S (k 2 +k 2 )M Ω ikm Γ )u = f Often: h k 1 but pollution effect: need h k 2??, h k 3/2?? Less dispersion: higher order FD or FE methods

9 Linear algebra problem weak form with absorption k 2 k 2 + iε, ( a ε (u, v) := u. v (k 2 + iε)uv ) ik uv Ω Γ = fv + gv Shifted Laplacian Ω Γ [Equivalently k 2 + iε (k + iρ) 2 ] Finite element discretization A ε u := (S (k 2 + iε)m Ω ikm Γ )u = f A ε somehow better behaved than A.

10 How bad things can be ε = 0 Solving Ax = f = 1 on unit square h k 3/2 Using GMRES (minimises residual in Krylov space: span{f, Af,... A k 1 f}) k n # GMRES > 1000

11 Preconditioning with A 1 ε Solve instead: A 1 ε Au = A 1 ε f. Theorem (with Martin Gander and Euan Spence) For Lipschitz star-shaped domains: If ɛ/k is sufficiently small then have GMRES converges independent of k. Proof: uses (high frequency) analysis of continuous problem. Warning: not a practical method (yet!)

12 Shifted Laplacian preconditioner ε = k Solving A 1 ε Ax = A 1 1 on unit square ε h k 3/2 k # GMRES

13 Shifted Laplacian preconditioner ε = k 3/2 Solving A 1 ε Ax = A 1 1 on unit square ε h k 3/2 k # GMRES

14 Shifted Laplacian preconditioner ε = k 2 Solving A 1 ε Ax = A 1 1 on unit square ε h k 3/2 k # GMRES

15 Exterior scattering problem with refinement h k 1, # GMRES 1 with diagonal scaling k ε = k ε = k 3/

16 A trapping domain k ε = k ε = k 3/2 10π/ π/ π/ π/ Betcke, Chandler-Wilde, IGG, Langdon, Lindner, 2010

17 Part II: Domain decomposition for A 1 ε Approximate by solves with A ε in subspaces: subdomains coarse grid Questions: Convergence, Scalability?

18 Classical additive Schwarz To solve a problem on a fine grid FE space S h Coarse space S H (here linear FE) on a coarse grid Subdomain spaces S i on subdomains Ω i, overlap δ Dirichlet BCs on subdomains

19 Classical additive Schwarz p/c for matrix C Approximation of C 1 : R T i C 1 i R i + R T HC 1 H R H i R i = restriction to S i, C i = R i CR T i R H = restriction to S H C H = R H CR T H Apply to A ε to get B 1 ε

20 Convergence results (work in progress) Theorem IGG, E. Spence, E. Vainikko, 2014 Consider solving B 1 ε A ε x = Bε 1 f n = number of GMRES iterates to achieve fixed accuracy provided n ( k 2 ε ) 4 ( ε ) 3/2 kh c k 2 So ε k 2 = robust method no pollution in coarse grid. Actually we want to solve Bε 1 Ax = Bε 1 f ε k? Numerical experiments: unit square, impedance BC

21 B 1 ε as preconditioner for A ε ε = k 2 h k 3/2, kh 1 k #GMRES Scale = Relative Coarse and subdomain problem size relative size of coarse and subdomain problems H = k 1 coarse grid subdomain Aggressive coarseing

22 B 1 ε as preconditioner for A ε ε = k 2 h k 3/2, kh k 0.1 k #GMRES Scale = Relative Coarse and subdomain problem size relative size of coarse and subdomain problems H = k 0.9 coarse grid subdomain Aggressive coarseing

23 B 1 ε as preconditioner for A ε ε = k 2 h k 3/2, kh k 0.2 k #GMRES Scale = Relative Coarse and subdomain problem size relative size of coarse and subdomain problems H = k 0.8 coarse grid subdomain Aggressive coarseing

24 B 1 ε as preconditioner for A ε and A ε = k h k 3/2, kh 1 k for A ε for A Scale = Relative Coarse and subdomain problem size relative size of coarse and subdomain problems H = k 1 coarse grid subdomain

25 When ε k Introduce more wavelike components: Impedance boundary condition on subdomains Hybrid restricted additive Schwarz

26 B 1 ε as preconditioner for A ε = k 20 grid points per wavelength, Hybrid RAS, Impedance subdomain problems kh k 0.5 k #GMRES Scale = Complexity (serial): n log n, 0 Relative Coarse and subdomain problem size relative size of coarse and subdomain problems, H= k 0.5 coarse grid subdomain h in 2D

27 B 1 ε as preconditioner for A ε = k 20 grid points per wavelength, Hybrid RAS, kh k 0.5 k GMRES, impedance GMRES, Dirichlet > > 600

28 Summary k and ɛ explicit analysis allows rigorous explanation of some empirical observations and formulation of new methods. When ɛ k, When ɛ k 2, A 1 ɛ B 1 ε is optimal preconditioner for A is optimal preconditioner for A ε When preconditioning A with B 1 ε, best choice is ε k Then BC of subdomain problems very important (impedance, PML,...). New analysis of Domain Decomposition methods.